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受欢迎的微积分 >

integral of (15-15x)/(1-sqrt(x))

解答

\int \frac{15 - 15 x}{1 - x} d x

解答

30 (\frac{1}{3} x^{\frac{3}{2}} + \frac{x}{2}) + C

求解步骤

\int \frac{15 - 15 x}{1 - x} d x

使用换元积分法

= \int 30 u (u + 1) d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 30 \cdot \int u (u + 1) d u

乘开

u (u + 1) : u^{2} + u

= 30 \cdot \int u^{2} + u d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 30 (\int u^{2} d u + \int u d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int u d u = \frac{u ^{2}}{2}

= 30 (\frac{u ^{3}}{3} + \frac{u ^{2}}{2})

u = x

代回

= 30 (\frac{( x ) ^{3}}{3} + \frac{( x ) ^{2}}{2})

化简

30 (\frac{( x ) ^{3}}{3} + \frac{( x ) ^{2}}{2}) : 30 (\frac{1}{3} x^{\frac{3}{2}} + \frac{x}{2})

= 30 (\frac{1}{3} x^{\frac{3}{2}} + \frac{x}{2})

解答补常数

= 30 (\frac{1}{3} x^{\frac{3}{2}} + \frac{x}{2}) + C

作图

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